Hyperstability of some functional equation on restricted domain‎: direct and fixed point methods

Abstract

The study of stability problems of functional equations was motivated by a question of S.M. Ulam asked in 1940. The first result giving answer to this question is due to D.H. Hyers. Subsequently, his result was extended and generalized in several ways.<br />We prove some hyperstability results for the equation <br /><br />g(ax+by)+g(cx+dy)=Ag(x)+Bg(y)<br /><br />on restricted domain. <br />Namely, we show, under some weak natural assumptions, that functions satisfying the above equation approximately (in some sense) must be actually solutions to it.